A NONLINEAR DYNAMICS PERSPECTIVE OF

WOLFRAM’S NEW KIND OF SCIENCE.

PART XII: PERIOD-3, PERIOD-6, AND

PERMUTIVE RULES

Leon O. Chua and Giovanni E. Pazienza*

Department of Electrical Engineering and Computer Sciences,

University of California at Berkeley,

Berkeley, CA 94720, USA * Cellular Sensory and Wave Computing Laboratory

Computer and Automation Research Institute

MTA-SZTAKI

Budapest, Hungary

Abstract

This 12th part of our Nonlinear Dynamics Perspective of Cellular Automata concludes a

series of three articles devoted to CA local rules having robust ω-limit orbits. Here we

consider only the two rules, 131 and 133 , constituting the third of the six groups in

which we classified the 1D binary Cellular Automata. Among the numerous theoretical

results contained in this article, we emphasize the complete characterization of the ω-

limit orbits, both robust and non-robust, of these two rules and the proof that period-3

and period-6 ω-limit orbits are dense for 131 and 133 , respectively. Furthermore, we

1

will also introduce the fundamental concepts of perfect period-T orbit sets and riddled

basins, and see how they emerge in rule 131 .

As stated in the title, we also focus on permutive rules, which have been introduced in a

previous installment of our series but never thoroughly studied. Indeed, we will review

some of the well-known properties of such rules, like the surjectivity, examining their

implications for finite and bi-infinite Cellular Automata.

Finally, we propose a new list of the 88 globally-independent local rules, which is slightly

different from the one we have used so far but has the great advantage of being via a

rigorous methodology and not an arbitrary choice. For the sake of completeness, we

display in the appendix the basin tree diagrams and the portraits of the ω-limit orbits of

the rules from this refined table which have not yet been reported in our previous articles.

Keywords: cellular automata; nonlinear dynamics; period-3 rules; period-6 rules; group 3

rules; permutive rules; surjective rules; surjectivity; dense orbits; perfect orbit sets; ω-

limit orbits; attractors; Isles of Eden; gardens of Eden; orbit concatenation; basin tree

diagrams; isomorphic basin trees; riddled basins of attraction; Wolfram.

2

1. List of the 88 minimal equivalence rules

The universe of one-dimensional cellular automata is small yet so rich in

nonlinear dynamics phenomena, as we have delineated over the past eleven episodes of

our chronicle: Part I [Chua et al., 2002], Part II [Chua et al., 2003], Part III [Chua et al.,

2004], Part IV [Chua et al., 2005a], Part V [Chua et al., 2005b], Part VI [Chua et al.,

2006], Part VII [Chua et al., 2007a], Part VIII [Chua et al., 2007b], Part IX [Chua et al.,

2008], Part X [Chua et al., 2009a], and Part XI [Chua et al., 2009b]. The final aim of our

research is to formalize the enormous amount of empirical results collected in Wolfram’s

monumental tome [Wolfram, 2002] via the rigorous theory of Nonlinear Dynamics.

This paper contains many relevant results, especially on the dynamics of period-3,

period-6, and permutive rules. However, we devote this first section to a fundamental

issue that will affect our future work. As proved in [Chua et al., 2004], the 256 one-

dimensional binary Cellular Automata can be partitioned into 88 global equivalence

classes, each containing 1, 2 or 4 elements. For the sake of simplicity, we have so far

chosen the rule with the lowest number as the representative specimen of the whole class.

For instance, in our discourse we often mentioned rule 60 , but never rule 102 , because

it is globally equivalent to 60 ; namely, ( )†102 60T= . It is important to emphasize

that this choice is arbitrary.

However, in [Chua et al., 2009b] we discussed the importance of the number 137 in

the history of physics, and we hence suggested to consider rule 137 , and not rule 110 ,

for representing the class of universal rules. This episode prompted us to reconsider our

original, rather arbitrary, procedure in selecting a representative rule for each equivalence

class and reflect on the possible alternatives to it. Indeed there is a better choice: we

3

select as representative of each of the 88 global equivalence classes the rule with the

smallest number of firing patterns [Chua et al., 2003] or, in other words, with fewer red

vertices in the Boolean cube. If two rules have the same number of firing patterns, pick

the smaller rule. From the perspective of training our monkey mascots, as depicted in our

cartoons [Chua, 2006], to raise a red flag whenever he sees a firing pattern, this implies a

minimal learning time: This procedure 1 allows us to include 137 among the 88

‘minimal’ rules, reported in Table 1a. Remarkably, only six rules change number with

respect to the previous list.

In [Chua et al., 2009b] we introduced the concept of quasi-global equivalence

which relates the space-time pattern of non-globally equivalent rules, proving that there

are six pairs of quasi-globally equivalent CA local rules. The list of the 82 minimal quasi-

equivalence rules is displayed in Table 1b. Here, we made two exceptions with respect to

the above selection procedure; namely, we included rule 170 instead of rule 15 , and

rule 204 instead of rule 51 . In both cases the reason has been the particular

importance that 170 and 204 have had throughout our work, specially since 170 is

exactly the Bernoulli shift when L . The Boolean cubes of the minimal and quasi-

minimal equivalence rules are displayed in Tables 2a and 2b, respectively. As already

mentioned, only six rules have been renumbered: two of them – rules

→∞

131 and 133 –

belong to Group 3 and are the subject of this article; two others – rules 129 and 161 –

belong to Group 5 (Hyper Bernoulli rules) and they are discussed in Appendix A; the last

two – rules 137 and 166 , belong to Group 6 (Complex Bernoulli rules) and they are

discussed in Appendix B.

To conclude this section, we present two tables summarizing the main properties –

such as the complexity index, the group number, the time-reversibility etc. – of the 256

local rules, in Table 3, and the 88 minimal rules, in Table 4.

1 Observe that a similar approach has been used in other works (see [Li & Packard, 1990]).

4

2. Basin Tree Diagrams, Omega-Limit Orbits and time-τ

characteristic function of rules from Group 3

In [Chua et al., 2009a] and [Chua et al., 2009b] we presented all basin tree

diagrams and portraits of ω-limit orbits of the local rules belonging to Groups 1 and 2,

respectively. Here, we do the same for the two rules of Group 3; namely, 131 and 133 .

First of all, we recall some fundamental concepts of our Nonlinear Dynamics Perspective

of Cellular Automata.

Each binary bit string is coded by an integer number equal to the decimal equivalent of

an L-bit binary bit string as follows:

( )0 1 2

0[ ... ] 2

II i

I ii

ix x x x n xβ −

=→ ∑ i (1)

where and 1I L− {0, 1}iβ ∈ .

Consequently, for each string length L, there are 2L distinct binary strings labeled from

n = 0 to . For computational purposes, such as in deriving the time-1 return

maps, it is necessary to convert a binary bit string to its associated real number

2Ln= −1

[0, 1)φ ∈

via the formula [Chua et al., 2005a]:

( 1)

02

Ii

ii

xφ − +

== ∑ (2)

where . 1I L−

The 2L binary strings for 3 along with their decimal number code n and the real

number

8L≤ ≤

φ are displayed in Table 7 of [Chua et al., 2009a].

For each rule N and length L, the basin tree diagrams give a complete catalog of the

evolution from all possible 2L initial binary bit strings of length L. In general, they are

organized into several isolated directed graphs. Each of these connected graphs is called

a basin tree in [Chua et al., 2006] for conciseness, even though some of these graphs are

not trees in the graph theoretic sense. Any connected component graph in the basin tree

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diagrams that has an empty set as its basin of attraction2 is called an Isle of Eden;

conversely, any connected component graph in the basin tree diagrams that has a non-

empty set as its basin of attraction is called an attractor [Chua et al., 2007a]. We usually

refer to both types of asymptotic orbits as ω-limit orbits [Chua et al., 2009a]. Note that

according to the classical definition [Birkhoff, 1927], an ω-limit orbit is always periodic

for finite L whereas the classical usage of the word orbit allows it to be the entire

trajectory (both transient and steady state). Hence, the ω-limit orbit excludes the transient

part of the orbit and, for finite L, it coincides with the periodic orbit.

2.1 Basin Tree Diagrams of rules from Group 3 In Tables 5a and 6a we present a gallery of the basin tree diagrams3 for rules

131 and 133 , respectively. They are organized into two Galleries, each identified by a

two digit number “N−m”, where the left digit N is identified with the local rule N ,

where { }131,133N ∈ , and the right digit m pertains to page m of Gallery N. The first

page (i.e., Gallery N-1) of each Gallery N displays the following relevant information:

1. Rule number N

2. Boolean cube of rule N

3. Explicit formula for generating the truth table of rule N , where

1 1, , {0,1}i i ix x x− + ∈

4. Truth table of rule N

5. Characteristic function 1Nχ of rule N

6. Sample set of time-1 return maps corresponding to different random bit strings

0 [0,1)φ ∈ , where the transient components have been deleted to avoid clutter. Each color

codes for an ω-limit orbit.

2 We define here the basin of attraction of a basin tree diagram as the collection of all “transient” (non-recurring) bit strings which converge to an attractor. 3 The basin tree diagrams for all 256 CA local rules are also available on the webpage http://sztaki.hu/~gpazienza/cellular_automata

6

Each Gallery N−m of rule N is composed of attractors and their basins of attraction,

coded in magenta, and Isles of Eden, coded in blue (a mnemonic for islands surrounded

by the blue sea). To estimate the robustness of each distinct attractor and each distinct

Isle of Eden, we simply calculate the ratio of the total number of bit strings, for each bit

length L, over the total number 2L of bit strings:

. " "2Lm

No of bit strings in attractor m or Isleof Eden mρ " " (3)

We will henceforth call the number mρ the robustness coefficient of the attractor “m”, or

the Isle of Eden “m” of the local rule N .

2.2 Portraits of the ω-Limit Orbits from rules of Group 3

Given any local rule N , and any initial bit string

0 1 2 1 0 0[ ... ] [Lx x x x 0,1)φ− → ∈ (4)

a fundamental problem is to determine the time-asymptotic behavior

0 1 2 1[ ... ] [L n nx x x x 0,1)φ− → ∈ (5)

as n → ∞. For finite L, the evolution must converge to a period-T orbit, where 2LT < .

For ease of future reference, all relevant qualitative properties of the ω-limit orbits

of rules 131 and 133 have been extracted, organized, and exhibited in two portraits in

Tables 5b and 6b, respectively. For each bit string length L, identified in column 1, only

distinct isomorphic4 attractors and Isles of Eden are displayed: they are identified in

column 2 by a red integer i. The information extracted from Tables 5a and 6a as

described above are collected in the columns 3 to 9. In addition, column 7 (Bernoulli

Parameters) shows the three relevant Bernoulli parameters (β, σ, τ) [Chua et al., 2005a]

for each rule N . Column 8 specifies the integer δmax defined as the distance from the

farthest Garden of Eden to the attractor of each basin tree diagram. Clearly, δmax = 0 for

4 A thorough discussion about isomorphic ω-limit orbits can be found in [Chua et al., 2009a].

7

all Isles of Eden. The last column 9 is devoted to the robustness coefficient for each

isomorphic attractor, or Isle of Eden. The robustness coefficient mρ for each orbit is

calculated and displayed in the form 2m L

number of bit stringskρ ⎛ ⎞= ×⎜ ⎟⎝ ⎠

so that the

number k of isomorphic copies of attractors, or Isles of Eden, can be identified directly.

Note that the sum of all mρ for each table is equal to 1.

Observe that in Tables 5a and 6a, we display all basin tree rules grouped by

genotype5 whereas in Table 5b and 6b, for the sake of briefness, we often grouped basin

trees according to their phenotype. For this reason, not always the values of seeds and

robustness correspond in these two tables.

2.3 Time-τ characteristic diagram of rules from Group 3

The first page of Table 5a displays the time-1 characteristic function of rule 131

extracted from Table 2 of [Chua et al., 2005a]. However, it is much more relevant

analyzing the time-3 characteristic function of this rule, since 131 has robust period-3

ω-limit orbits. Figure 1 is an empirical evidence of this phenomenon, because most of the

points of the time-3 characteristic function lie on the diagonal, and hence they correspond

to period-3 ω-limit orbits, whereas only a few do not lie on the diagonal, and hence they

correspond to transients and non-robust ω-limit orbits. From Fig. 2, we observe that a

great percentage of the 2L bit strings belongs to period-3 ω-limit orbits for rule 131 even

for low values of L, such as . 15L ≥

Similarly, the first page of Table 6a displays the time-1 characteristic function of

rule 133 , also extracted from Table 2 of [Chua et al., 2005a]. For this rule, it is much

more relevant analyzing its time-6 characteristic function, since 133 has robust period-6

ω-limit orbits. Figure 3 is an empirical evidence of this phenomenon, since most of the

points of the time-6 characteristic function lie on the diagonal, and hence they correspond

to period-6 ω-limit orbits, whereas only a few do not lie on the diagonal, and hence they

5 A genotype characterizes a set of isomorphic basin trees; if non-isomorphic basin trees have the same diagraph representation, then they are said to have the same phenotype [Chua et al., 2009a].

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correspond to transients and non-robust ω-limit orbits. The existence of robust period-6

ω-limit orbits is even more evident is Fig. 4, in which we focused on a smaller portion of

the axis and with an higher value of L. From Fig. 5, we observe that a great

percentage of the 2

[ )0.5,1φ ∈

L bit strings belongs to period-6 ω-limit orbits for rule 133 even for

low values of L, such as . 27L ≥

3. Robust ω-limit orbits of rules from Group 3

Rule 131 and rule 133 have been included in Group 3 because they have robust

period-3 and period-6 orbits, respectively, as found numerically in [Chua et al., 2007b].

We proposed this classification after verifying that several dozens of randomly-chosen

long (400 bits) bit strings indeed belonged to period-T ω-limit orbits, where T is equal to

3 for rule 131 and 6 for rule 133 . This empirical approach has in fact a mathematical

justification: if we let q denote the fraction of non-robust period-T orbits for a rule from

Group 3 (either rule 131 or rule 133 ) then the probability p that m randomly-chosen bit

strings belong to a non-robust orbit is p=qm; for instance, for q=0.01 (which means that

only one bit string out of 100 belongs to a non-robust orbit) and m=50 (we choose

randomly 50 bit strings), the probability that all of these iterates converge to a non-robust

orbit . Therefore, the probability that 10010mp q −= = 131 or 133 are misclassified is

extremely low.

Even though this methodology is effective, our ‘Nonlinear dynamics perspective

of Cellular Automata’ is based on rigorous proofs and firm results, and hence we cannot

rely exclusively on computer simulations. For this reason, we will present in Sec. 3.2 the

formal proofs concerning the robust ω-limit orbits for rules 131 and 133 , after

reviewing in Sec. 3.1 some basic definitions. We recall that a similar rigorous approach

was used in [Chua et al., 2009a] to prove that Group 1 rules have robust period-1 ω-limit

orbits, and in [Chua et al., 2009b] to prove that Group 2 rules have robust period-2 ω-

limit orbits.

9

3.1. Definitions and notation We start our analysis by formalizing the concept of period-T robustness as

follows:

Definition 3.1.1. Robust period-T ω-limit orbits

A local rule has robust period-T ω-limit orbits if the probability for a random bit string of

finite length L to coverge to a period-T ω-limit orbit6 tends to 1 when . L →∞

This definition does not exclude a period-T rule from having other ω-limit orbits

(attractors and Isles of Eden) with period different from T; however, if such ω-limit

orbits exist, they are non-robust, in the sense that they appear only for some values of L

and their basins of attraction are composed of a limited number of bit strings satisfying

very specific conditions. Note that this definition extends those given in [Chua et al.,

2009a] and [Chua et al., 2009b] for period-1 and period-2 rules, respectively. Definition

3.1.1 implies that a period-T rule is characterized by the fact that for a given L most of

the 2L bit strings belong to the basin of attraction of period-T ω-limit orbits (attractors

and Isles of Eden).

Also, we would like to recall a definition and a fundamental theorem about the

periodicity of the single bits of a generic bit string [Chua et al., 2009b]: nx

Definition 3.1.2. Period-T bit

Given a bit string belonging to a periodic orbit and denoting by nx nix the bit of at

position i, the bit

nx

nix has period T if n T n

i ix x+ = .

In other words, a bit has period T if it repeats in the same position after every T iterations.

Recursively, we find that if a bit has period T, then n kT ni ix x+ = , k∀ ∈ .

6 For finite L, the ω-limit orbit coincides with the periodic orbit associated with either an attractor, or an isle of Eden. However, for , a ω-limit orbit need not be periodic. L = ∞

10

Theorem 3.1.1.

The period of a bit string ( 0 1n n n n

I )x x x=x … is equal to the least common multiple of the

periods of its bits.

As a corollary, if all bits of a bit string have the same minimal periodicity T, then also the

bit string has period T; however, the converse is not true, because the fact that a bit string

has period T does not imply that all its bits have period T. Indeed, some bits may have

shorter periods in view of some local symmetry within the bit string.

We would like to conclude this section by giving a few details about the notation

used here for consistency with [Chua et al., 2009a] and [Chua et al., 2009b], and hence

already familiar to the readers of our previous works.

In the following, we use the expression ‘isolated 0’ (resp., ‘isolated 1’) to indicate a bit 0

(resp., 1) whose left and right neighbors are 1 (resp., 0), and the expression ‘runs of k 0’

(resp., ‘runs of k 1’) to indicate k consecutive bits 0 (resp., 1). Moreover, the next section

contains numerous figures showing the predecessors of a given pattern under rule N .

Such pattern is always on the last row – labeled Pattern – whereas its possible

predecessors are on the other rows, labeled Case I, Case II, etc. These bit strings are

created bit by bit starting from the left and trying all possible valid possibilities; we

employ the symbol X when no bit in that position is possible under rule N . These

figures allow us to prove that some sequences cannot appear in the periodic orbits of

131 and 133 . As an example, we give a detailed and thorough explanation of Fig. 7, in

order to help the readers to follow our proofs.

Example 3.1.3 – Interpretation of Figure 7

Our goal is to prove that the bit strings belonging to the ω-limit orbits of rule

131 cannot contain the pattern 101101. For example, it would be sufficient to show that

can appear only in Gardens of Eden, or, in other words, it has no predecessors.

We can denote the generic predecessor of 101101 by the bit string

101101

0 1 2 3 4 5 6 7x x x x x x x x ;

11

hence, 1310 1 2 3 4 5 6 7 101101x x x x x x x x ⎯⎯→ , where the symbol 131⎯⎯→ means ‘generated in

one iteration under rule 131 ’. Consequently, we would like to find { }0 1 7, , ,x x x… such

that

0 1 2 1x x x → (6.1)

1 2 3 0x x x → (6.2)

2 3 4 1x x x → (6.3)

3 4 5 1x x x → (6.4)

4 5 6 0x x x → (6.5)

5 6 7 1x x x → (6.6)

Our task is proving that no combination of { }0 1 7, , ,x x x… can give 101101.

From Table 7, we can also observe that the only firing patterns of 131 are ,

, and 111. Since

000

001 0 1 2x x x must be a firing pattern (see Eq. 6.1), it is sufficient to

perform a case study of the three cases: a) ( ) ( )0 1 2 111x x x = ; b) ( ) ( )0 1 2 001x x x = ; c)

. ( ) (000= )0 1 2x x x

)

In Case I, depicted in the first row of Fig. 7, we suppose that in the first three

columns we have ( ) ; therefore, (0 1 2 111x x x = 1313 4 5 6 7111 101101x x x x x ⎯⎯→ . We can now

focus on 3x : since and 1 1x = 2 1x = , 3x must be 0 to meet the condition of Eq. (6.2),

because 111 is a firing pattern. Therefore, 1314 5 6 71110 101101x x x x ⎯⎯→ . It is easy to

observe that we have obtained a contradiction, because neither 4 0x = nor can

meet the condition of Eq. (6.3) since both 100 and 101 are quenching patterns; hence,

. We will henceforth denote a disallowed case with a red X.

4 1x =

( ) (0 1 2 111x x x )≠

Let us suppose next that ( ) ( )0 1 2 001x x x = , i.e., 1313 4 5 6 7001 101101x x x x x ⎯⎯→ . We

focus on 3x : since and 1 0x = 2 1x = , the condition of Eq. (6.2) is always met, because

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both 010 and are quenching patterns. We analyze the case 011 3 1x = in Case II, and the

case in Case III. 3 0x =

Case II is depicted in the second row of Fig. 7, and, according to the assumptions

above, and 3 1x = 1314 5 6 70011 101101x x x x ⎯⎯→ . Observe that, because of Eqs. (6.3) and

(6.4), both 4x and 5x must be 1; hence, 1316 7001111 101101x x ⎯⎯→ . We can find the

value of 6x from Eq. (6.5): since 4 5 6x x x must be a quenching pattern and ,

then , because 111 is a firing pattern; hence,

4 5 1x x= =

6 0x = 13170011110 101101x ⎯⎯→ . Again, we

have obtained a contradiction, because neither 7 0x = nor 7 1x = can meet the condition

of Eq. (6.6), because both 100 and 101 are quenching patterns.

Case III is depicted in the third row of Fig. 7, and, according to the assumptions

above, and 3 0x = 1314 5 6 70010 101101x x x x ⎯⎯→ . Once more, we have obtained a

contradiction, because neither 4 0x = nor 4 1x = can meet the condition of Eq. (6.3) since

both 100 and 101 are quenching patterns. Therefore, ( ) ( )0 1 2 001x x x ≠ .

The last case to analyze, Case IV depicted in the fourth row of Fig. 7, is the one in

which ( ) , i.e., (0 1 2 000x x x = ) 1313 4 5 6 7000 101101x x x x x ⎯⎯→ . Also in this case, we obtain a

contradiction, because neither 3 0x = nor 3 1x = can meet the condition of Eq. (6.2) since

both 000 and are firing patterns. Therefore, 001 ( ) ( )0 1 2 000x x x ≠

Thanks to this exhaustive analysis, we have proved that no combination of

{ }0 1 7, , ,x x x… can give rise to 101101, which can hence be contained only in some

Gardens of Eden.

A similar approach is repeated for all figures of this section. Clearly, a thorough

description of all of them would be impractical, but the reader can try to repeat the proofs

in order to get familiar with this reductio ad absurdum methodology.

3.2. Rigorous results about the robust ω-limit orbits of rules 131 and 133

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In this section, we prove that rule 131 and rule 133 have robust period-3 and

period-6 ω-limit orbits, respectively. The behavior of these two local rules, both

belonging to Group 3, is more complex than those of the first two groups; for example,

they have very long transients and many non-robust ω-limit orbits. Consequently, the

proofs are also more complicated, even though they require only the knowledge of the

truth tables of rules 131 and 133 , which are shown in Table 7.

We start with the theorem concerning rule 131 and its globally-equivalent local rules.

Theorem 3.2.1. Period-3 rules

Rule 131 and its globally-equivalent local rules have robust period-3 ω-limit orbits.

Proof

This proof is particularly complex and hence, for the reader’s convenience, we

divided it into three parts: in the first part, we prove that the bit strings belonging to the

ω-limit orbits of rule 131 cannot contain an arbitrary number of consecutive 0s or 1s; in

the second part, we demonstrate that all ω-limit orbits are either period-3 or στ-Bernoulli

shift with σ=-1 and τ=2; in the third part, we show that only period-3 ω-limit orbits are

robust.

Part 1 – Form of the bit strings belonging to the ω-limit orbits of rule 131

The final aim of this part is finding the maximum number of consecutive 0s or 1s

that can be contained in the bit strings belonging to ω-limit orbits. For the moment, we do

not consider the bit strings 00...0 and 11..1 which will be analyzed in the second part of

the proof.

As for the runs of 0s, we observe that a bit string belonging to a periodic orbit

cannot include a run of seven or more 0s because all its valid predecessors have to

contain either the bit string 101101 or the bit string 10101, as proved in Fig. 6. However,

these two bit strings can appear only in some Gardens of Eden, as proved in Figs. 7 and 8,

respectively. Similarly, a run of six 0s cannot be part of a bit string of a periodic orbit

14

since it is transformed, after four iterations, into the bit string 001100 (see Figs. 9) which

belongs either to a bit string of the transient regime or to a στ-Bernoulli shift ω-limit orbit

with σ=-1 and τ=2, as proved later in this section (refer also to Fig. 14a). Therefore, the

bit strings of the ω-limit orbits of rule 131 cannot contain more than five consecutive 0s.

As for the runs of 1s, Fig. 5 shows that a run of five or more 1s has two possible

valid predecessors: either a run of k+1 0s or a run of k+2 1s. Therefore, the longest run of

1s that can belong to a bit string of an ω-limit orbits is 1111, because of the restriction of

the consecutive number of 0s. Consequently, the bit strings belonging to ω-limit orbits of

rule 131 cannot contain more than four consecutive 1s.

These considerations limit the number of consecutive 0s and 1s that can appear in

periodic orbits (no more than five consecutive 0s or four consecutive 1s), and hence we

can perform a thorough case-by-case study. Moreover, since runs of k 0s generate runs of

k-1 1s in one iteration, we can restrict our analysis to only four cases; namely, the

patterns 011110, 01110, 0110, and 010. The only possibility left out is the pattern 101,

which will be analyzed separately in Part 2 below.

Part 2 – Finding the periodicity of the ω-limit orbits of rule 131

The final aim of this part is proving that all possible cases that can arise in ω-limit

orbits, which have been identified in the previous part, are either period-3 or στ-Bernoulli

shift with σ=-1 and τ=2.

First, we analyze a run of four 1s (i.e., pattern 011110). Here, we consider eight

different situations, corresponding to all possible combinations of the three bits on the left

of the pattern 011110. We can find that there are three possible outcomes: a) if these

three bits are 000, 010 or 110 then the only ω-limit orbit to which the resulting pattern

can belong to must be period-3, as shown in Figs. 11a and 11b; b) if these three bits are

100 then we are in a transient regime because, as shown in Fig. 11c, after three iterations

we find the pattern 0111110 which cannot be included in any bit string of a periodic orbit

(refer to Fig. 9); c) if these three bits are 001, 101, 011, or 111 we obtain a bit string

including the pattern 10111 which, as proved in Fig. 12, can be included only in a

transient orbit.

15

Second, we analyze a run of three 1s (i.e., pattern 01110). Also in this case we

have to analyze eight different situations, corresponding to all possible combinations of

the three bits on the left of the pattern 01110. It is possible to observe that there are five

possible outcomes: a) if the three bits are 000 then the only ω-limit orbit to which the

resulting pattern can belong to must be period-3, as shown in Fig. 13a; b) when the three

bits are 010, 110 or 100, after three iterations we obtain either the bit string 01110 again

(see Fig. 13b) or the bit string 011110 (see Fig. 13c) which, if belonging to an ω-limit

orbit, is period-3, as proved before; c) if the three bits are 001 then we obtain after three

iterations the pattern 001100 (see Figs. 13d) which belongs either to a transient regime or

to a στ-Bernoulli shift ω-limit orbit with σ=-1 and τ=2, as proved later in this section

(refer also to Fig. 14a); d) if the three bits are either 011 or 111 then the resulting bit

string would contain the pattern 10111, which cannot be included in any bit string of a

periodic orbit (refer to Fig. 12); e) when the three bits are 101 the resulting bit string

would contain the pattern 10101, which cannot be included in any bit string of a periodic

orbit (refer to Fig. 8).

Third, we analyze a run of two 1s (i.e., pattern 0110) which are tackled by

considering the nearest neighbors (one on the left and one on the right) of the pattern

0110 and examining the three possible outcomes: a) if both the left and the right neighbor

are 0, then we obtain a στ-Bernoulli shift ω-limit orbit with σ=-1 and τ=2, as shown in

Fig. 14a; b) when only one of the two neighbors is 1, as in Figs. 14b and 14c, after two

iterations we obtain a run of four 1s (observe that, as shown in Fig. 14d, it is not possible

to obtain only three consecutive 1s) which belongs either to a transient or to a period-3 ω-

limit orbit, as proved earlier in this section; c) the case in which both neighbors are 1,

corresponding to the pattern 101101, can occur only in a Garden of Eden (refer to Fig. 7).

Fourth, we analyze what happens to isolated 1s – pattern 010 – through the usual

method, i.e., by considering all possible combinations of the three bits on the left of the

pattern 010 and examining the four possible outcomes: a) when the three bits are 000,

010, 110, 001 or 111 (see Figs. 15a, 15b, 15c and 15d), the pattern can only belong to a

period-3 ω-limit orbit (or, of course, to a transient); b) if the three bits are 100 (see Fig.

15e) then we obtain after one iteration the pattern 001100 which belongs either to a

transient regime or to a period-3 ω-limit orbit, as proved earlier in this section; c) when

16

the three bits are 011 (see Fig. 15f) we obtain a run of four 1s which belongs either to a

transient regime or to a period-3 ω-limit orbit, as proved earlier in this section; d) when

the three bits are 101 the resulting bit string would contain the pattern 10101 which

cannot be included in any bit string of a periodic orbit (refer to Fig. 8).

Fifth, in Fig. 16 we show that the pattern 101 is transformed in two iterations into

one of the patterns already studied.

We conclude this part with an important observation. At the beginning of Part 1

above, we specified that we were not considering the bit strings 00...0 and 11...1: the

reason is that they converge to a period-1 ω-limit orbit (11...1→11...1) which has a

reduced basin of attraction since it includes only bit strings having some very special

spatial symmetry. Moreover, such orbit is at the same time period-1, period-3 and στ-

Bernoulli shift ω-limit orbit with σ=-1 and τ=2, and hence we do not consider it as a

separate case.

This proves all bit strings for any finite L belong to two kinds of orbits: either

period-3 or στ-Bernoulli shift with σ=-1 and τ=2.

Part 3 – Robustness of the ω-limit orbits of rule 131

In the previous part, we proved that all ω-limit orbits of rule 131 are either

period-3 or στ-Bernoulli shift. Now, we will show that only the former are robust.

Looking carefully at the case study performed in Part 2, we find that there are two

possible period-3 patterns: the first one, which we indicate with the symbol P1, consists of

a run of three 1s (see Fig. 13a) and its space-time pattern is depicted in Fig. 17; the

second one, which we indicate with the symbol P2, consists of a run of four 1s (see Figs.

11a and 11b) and its space-time pattern is depicted in Fig. 18. Also, there is only one στ-

Bernoulli shift pattern, which we indicate with the symbol B, consisting of a run of two

1s and whose space-time pattern is depicted in Fig. 19. Observe that patterns containing

isolated 1s can be period-3 (see Figs. 15c and 15d), στ-Bernoulli with σ=-1 and τ=2 (see

Fig. 15e), or even both period-3 and στ-Bernoulli shift at the same time (see Figs. 15a and

15b).

17

Now, we need to see how the στ-Bernoulli shift pattern B interacts with the

period-3 patterns P1 and P2. Through an exhaustive case study, not reported here to avoid

numerous and tedious examples, we found that there are only two possible scenarios,

illustrated in Figs. 20 and 21. In the first case, the Bernoulli shift pattern B ‘hits’ the

period-3 pattern P1, generating a period-3 pattern P2. In the second case, the Bernoulli

shift pattern B ‘hits’ the period-3 pattern P2, generating another Bernoulli shift pattern B.

Thanks to this observation, we can draw a fundamental conclusion: if the number of

patterns B generated in the transient regime is greater that the one of patterns P1, then the

resulting ω-limit orbit is a στ-Bernoulli shift with σ=-1 and τ=2; otherwise, the ω-limit

orbit is period-3. Therefore, we need to focus now on what happens in the transient

regimes of rule 131 .

Unfortunately, this rule has particularly long transients, which are impractical to

analyze case by case (especially because their lengths depend on L); this makes it

difficult to find rigorous conclusions. However, we can here give an intuitive argument

that explains the results found experimentally, and hence helps us to prove the theorem.

Let us consider a bit string of length L, where L is possibly very long, belonging to a

period-3 ω-limit orbit; consequently, the number of patterns P1 generated in the transient

regime is greater than or equal to that of patterns B. Now, we add three arbitrary bits on

the right, as shown in Fig. 22, and we analyze what the possible outcomes are. If the bit

string is long enough, the number of P1 and B patterns generated in the ‘central part’ of

the bit string, shown in dark grey in Fig. 22, will not change (or change slightly), whereas

new B and P1 patterns will be generate by the three added bits and their two neighbors (in

light grey in Fig. 22). If we make an exhaustive analysis of the behavior of the 25=32

possible 5-bit patterns, shown in Fig. 23, we find that in 22 cases out of 32 the new

pattern is also period-3 while only in 14 cases out of 22 the patterns generated are στ-

Bernoulli shifts. Therefore, if the initial bit string belongs to a period-3 orbit, then in the

majority of cases we still obtain a period-3 orbit when we increase such bit strings by

three bits; however, if the original bit string belongs to a στ-Bernoulli shift orbits, a new

equilibrium between B and P1 patterns can be created, and a new period-3 bit string may

still be generated as a result.

18

Summary and Conclusion

In the first part of the proof, we focused on the form of the bit strings belonging to

the ω-limit orbits of 131 , finding that they cannot contain more than five consecutive 0s

or four consecutive 1s. Thanks to this result, which allowed us to perform an exhaustive

case analysis on the different situations that can arise, we proved, in the second part, that

all ω-limit orbits of 131 are either period-3 or στ-Bernoulli shifts. Moreover, as shown

in the third part, the percentage of period-3 ω-limit orbits of rule 131 is always

increasing over the total as L tends to infinity, which is exactly the definition of robust ω-

limit orbits.

In conclusion, rule 131 has robust period-3 ω-limit orbits.

The robustness of the period-3 ω-limit orbits can be observed even better in Fig.

24, in which we analyze separately the three cases for 3L m k= + , k ∈ : i)

; ii) ; iii) 0 mod1m ≡ 0 mod 2m ≡ 0 mod3m ≡ .

We can now focus on rule 133 and its globally-equivalent local rules.

Theorem 3.2.2. Period-6 rules

Rule 133 and its globally-equivalent local rules have robust period-6 ω-limit orbits.

Proof

Also this proof is very complex and we divided it into three parts: in the first part,

we prove that the bit strings belonging to ω-limit orbits for rule 133 cannot contain an

arbitrary number of consecutive 0s or 1s; in the second part, we observe that all robust ω-

limit orbits have a common feature (i.e., they have what we will dub ‘walls’); in the third

part, we demonstrate that all bits of the bit strings belonging to robust ω-limit orbits can

19

be exclusively period-1, period-2, or period-3. The conclusion that all robust ω-limit

orbits of rule 133 have period-6 follows from Theorem 3.1.1.

Part 1 – Form of the bit strings belonging to the ω-limit orbits of rule 133

The final aim of this part is finding the maximum number of consecutive 0s or 1s

that can be contained in the bit strings belonging to ω-limit orbits.

As for the runs of 0s, we observe that a run of three 0s is transformed, in one

iteration, into a stable pattern (i.e., not disrupted during the evolution of the Cellular

Automaton) containing an isolated 1 (see Fig. 25); hence, it cannot be part of a bit string

belonging to an ω-limit orbit. A similar argument can lead to the conclusion that all

patterns of k consecutive 0s, k odd, cannot be part of a bit string belonging to an ω-limit

orbit, as shown in Fig. 26 for k=5.

As for the runs of 1s, if k is odd, k greater than or equal to 3, then the only valid

predecessor is a run of k+2 1s (see Fig. 27), but this implies that we are in a transient

regime and not in an ω-limit orbit.

We conclude this first part of the proof by emphasizing the two main results

obtained so far: first, the bit strings belonging to ω-limit orbits cannot contain runs of k

consecutive 0s or 1s, where k is odd and greater than 1; second, isolated 1s (i.e., the

pattern 010) are stable (i.e., period-1), which means that the ‘walls’ formed by an isolated

1 cannot be disrupted during the evolution of the Cellular Automaton.

Part 2 – Robust ω-limit orbits of rule 133

In the previous part we saw that the sequence 010 forms a ‘wall’ in the space-time

pattern, because 010 . However, since the probability that a randomly-chosen bit

string of length L includes the pattern 010 tends to 1 as L tends to infinity, we can state

that any bit string belonging to a robust ω-limit orbit contains at least one such a ‘walls’.

This implies that no σ

010→

τ-Bernoulli shift ω-limit orbit can be robust.

Moreover, for finite L, the ‘wall’ limits the dynamic of the Cellular Automaton on

both sides, due to the periodic boundary conditions. Our strategy will then be to analyze

20

thoroughly what dynamics arise between two ‘walls’ and combine them through

Theorem 3.1.1.

An important observation is that rule 133 is bilateral; i.e., ( )†133 133T= .

Consequently, we can analyze only what happens at the right side of a ‘wall’, because the

same behavior occurs on the left side too.

Part 3 – Periods of the ω-limit orbits of rule 133

The final part of this proof is devoted to explore the periodicity of robust ω-limit

orbit of rule 133 . We will analyze all possible cases described in Part 1 in the

framework proposed in Part 2 (i.e., right side of a ‘wall’).

First, Fig. 28 shows that a run of six or more consecutive 1s on the right of a

‘wall’ results in a run of three 0s after two iterations, which form a new ‘wall’ (refer to

Fig. 25). Therefore, a run of k 0s, where k is greater than 5, cannot be in an ω-limit orbit.

Second, Fig. 29a shows that a run of four 1s gives, after three iterations, a run of

three 1s: since five consecutive 1s cannot be generated in a periodic orbit, as proved

before, we need to test whether either a run of three 1s or a run of four 1s can be

produced. The first case is excluded by the considerations made in the first part of this

proof, whereas the second case is confirmed by Fig. 29c: to sum up, a run of four 1s can

be part of a bit string belonging to a period-3 ω-limit orbit.

Third, Fig. 30a shows that a run of two 1s is transformed, after two iterations, into

a run of at least two 1s; therefore, two alternatives are possible: the first possibility is that

the pattern produced is 0110, which means that the run of two 1s is a period-2 ω-limit

orbit, as illustrated in Fig. 30b; the second possibility is that the pattern produced is

011110, which means that the run of two 1s was either in a transient regime or in the

basin of attraction of a period-3 ω-limit orbit.

To sum up, any run of k consecutive 0s, k even, must be in a transient, a period-2

ω-limit orbit, or a period-3 ω-limit orbit.

Now, let us analyze how runs of k consecutive 0s, k even, evolve. In Fig. 31 we

observe that a run of four consecutive 0s is transformed, in one iteration, into a pattern

containing at least two consecutive 1s which, as seen previously, can be in a transient

21

regime, a period-2 ω-limit orbit, or a period-3 ω-limit orbit. Figure 32a analyzes a run of

two consecutive 0s which evolves, in one iteration, in a pattern containing at least two

consecutive 0s. This implies that such pattern is either period-2, as depicted in Fig. 32b,

or is transformed in a run of four 0s.

In conclusion, all patterns on the right (and consequently on the left, due to the

bilateral property of rule 133 ) of a ‘wall’ can be only period-1 (Fig. 32b), period-2 (Fig.

30b) or period-3 (Fig. 29b).

Finally, we have to check what happens on the right of these patterns. If we find

that only the three patterns mentioned before can arise, then our proof is concluded.

Through an exhaustive study of all possible cases, we found that the only way in which

the pattern in Fig. 29b can be in an ω-limit orbit is through the scheme of Fig. 33 (and the

resulting pattern is still period-3), whereas for the pattern in Fig. 30b there are two

possible cases, in Figs. 34 and 35 (and the resulting pattern is still period-2 in both cases).

This proves that all bit strings belonging to ω-limit orbit and having at least one isolated

1s, corresponding to the robust case, are period-6.

Summary and Conclusion

In the first part of this proof, we focused on the form of the bit strings belonging

to the ω-limit orbits of rule 133 , finding that they cannot contain runs of k consecutive

0s or 1s, where k is odd and greater than 1. Then, in the second part we presented some

fundamental properties of the robust ω-limit orbits of rule 133 . These considerations

have led to the proof, presented in the third part, that all bits of the bit strings belonging

to robust ω-limit orbits can be exclusively period-1, period-2, or period-3.

In conclusion, from Theorem 3.1.1 it follows that all robust ω-limit orbits

(attractors or Isles of Eden) of rule 133 have period 6.

As mentioned in the proof, this statement is certainly true for strings containing

the pattern 010 (which we called the ‘wall’) and hence for the robust case; however, non-

robust orbits may, or may not, be period-6. As an example, we show in Fig. 36 the case

22

of a period-6 ω-limit orbit for L=10, and in Fig. 37 the case of a period-14 ω-limit orbit

for L=14.

Theorems 3.2.1 and 3.2.2 follow a series of rigorous results concerning the

classification of CA local rules: Group 1 rules were analyzed in [Chua et al., 2009a],

Group 2 rules in [Chua et al., 2009b], Group 5 rules in [Chua et al., 2007a], and Group 6

rules in [Chua et al., 2007b]. We will conclude this discourse in our next paper, in which

we will focus on the rules belonging to Group 4 (Bernoulli στ-shift rules).

3.3. Dense and perfect ω-limit orbits, Riddled basins of attraction

Thanks to our incisive research on the dynamics of rules 131 and 133 , we are

able to characterize the distribution of their ω-limit orbits7. We start by presenting the

following theorem concerning the concatenation of an ω-limit orbit of a generic rule N :

Theorem 3.3.1 Concatenation of ω-limit orbits

Given a bit string ( 0 1 1L )x x x −=x … of length L representing an ω-limit orbit of rule ( )LΓ

N , the bit string , obtained by concatenating

m times , belongs to the ω-limit orbit

( ) (m times

0 1 1 0 1 1Lx x x x x x−′ = =x x x x… … … … )L−

x ( )L′Γ obtained by concatenating m times ( )LΓ .

Proof

It is a direct consequence of Theorem 3.3.1 (ω-Limit orbits generation algorithm) proved

in [Chua et al., 2009a], when L L′ ′′= and ( ) ( )L L′ ′ ′′ ′Γ ≡ Γ ′

.

Corollary 3.3.1 Concatenation of ω-limit orbits

7 Note that our definition of orbit does not require that it is periodic. This is consistent with standard usage in dynamical systems [Hasselblatt and Katek, 2003]. An orbit is just the trajectory from any initial condition, and is not periodic unless the initial state falls on a periodic orbit.

23

Given a bit string ( 0 1 1L )x x x −=x … of length L belonging to an period-T ω-limit orbit of

rule8 N , the bit string of length , obtained

by concatenating the bit string m times is also a period-T ω-limit orbit of rule

( ) (m times

0 1 1 0 1 1Lx x x x x x−′ = =x x x x… … … … )L− mL

x N .

The application of this corollary is evident in the examples of Figs. 38 and 39 for rules

131 and 133 , respectively.

In order to present some fundamental definitions and theorems, we need to introduce the

following notation: denotes the bit string space of all bit strings of length L; ( )LΣ 2L

( )T Lx denotes a period-T bit string of rule N from bit string space ( )LΣ ; ( )T pLx

denotes the period-T bit string of rule N derived by concatenation of ( )T Lx by p

multiples, where p is an integer. Now, we can give the definition of dense period-T

orbits:

Definition 3.3.1 Dense period-T orbits

The period-T orbits of rule N is said to be dense iff for any 0ε > , and any period-T

orbit , an integer such that for all ( )( )T L L∈Σx ∃ m p m≥ , there is another period-T

orbit ( ) ( )T pL p∈Σy L such that ( ) ( )T TpL pL ε− <x y .

Figure 40 gives a graphical representation of this definition. If we concatenate m times a

bit string of length L, we obtain a bit string of length mL. Saying that

another bit string must be closer than

( )T Lx ( )T mLx

( )T mLy ε to is equivalent to saying that

the first t bits of and are the same, where

( )T mLx

( )T mLy ( )T mLx 2logt ε= − ⎡ ⎤⎢ ⎥ . Those bits

beyond tx and can be considered as a ‘noise tail’. This concept is expressed in the

following Proposition:

ty

8 We follow the notation in [Chua et al., 2009a] where ( )LΓ denotes any ω-limit orbit of rule N , of length L.

24

Proposition 3.3.1 Noise-tail bounding estimate

The distance between any two bit strings whose first n bits are identical is less that 1

12n+ .

Example 3.3.1

Let x and y be two strings with 20L = such that ( )11011010111010010101=x and

. Since ( )11011010111101001011=y [ ], 0,10i ix y i= ∈ and 11 11x y≠ , then the distance

between this two bits is less than . Indeed, by using formula (2), we

find that

122 0.000244141− =

120.000173569 2x yφ φ −− = < .

Therefore, Definition 3.3.1 implies that for any 0ε > (and hence for any position t ),

given any period-T orbit and concatenating it times to generate the bit string

, where , we can create another period-T bit string where the first

bits of and coincide.

( )T Lx m

(T mLx ) mL t> ( )T mLy

t ( )T mLx ( )T mLy

If all period-T ω-limit orbits of rule N are dense for all possible periods, then

such set of ω-limit orbits is said to be perfect, according to the following definition:

Definition 3.3.2 Perfect period-T orbit set

The set { }:T TX x x x= = of all period-T orbits, for all possible periods, is said to be

perfect iff each T Tx X∈ is dense.

Observe that for the limit when , this definition is similar to the notion of a perfect

metric space [Hasselblatt and Katok, 2003].

L →∞

Now, we can analyze whether the robust orbits for rules 131 and 133 are dense.

Theorem 3.3.2 Dense period-3 orbits of rule 131

25

Rule 131 has a dense set of period-3 orbits.

Proof

In Theorem 3.2.1 it was proved that a bit string is period-3 under rule ( )T Lx 131

if, and only if, the number of patterns PnP1 1 generated in the transient regime is greater

than or equal to the number of patterns B. Therefore, we need to analyze how these

numbers change when we concatenate strings.

nB

Let us suppose that is a period-3 bit string; Theorem 3.2.1 implies that

. As we explained above and shown in Fig. 40, the value of

3 ( )Lx

n n≥P1 B ε determines the

position of the bit 1tx + , which is the starting point of the noise tail. According to

Definition 3.3.1, we want to find a period-T bit string of length in which the

first bits coincide with those of , obtained by concatenating m times ;

however, this implies that has a noise tail of length

(mLy )

)

mL

1t + 3 (mLx 3 ( )Lx

( )mLy 1mL t− − , where is

arbitrary. Moreover, since all bits of the noise tail of are also arbitrary, we can

always set them in such a way that the number of patterns P

m

( )mLy

nP1 1 is greater than or equal

to the number of patterns B, and hence the bit string has also period-3,

.

nB ( )mLy

3( ) ( )mL mL≡y y

In conclusion, rule 131 has a dense set of period-3 orbits.

Theorem 3.3.3 Dense period-T orbits of rule 131

Rule 131 is perfect.

Proof

Since rule 131 has only two types of periodic orbits, period-3 and στ-Bernoulli

shift orbits with σ=-1 and τ=2 (see Theorem 3.2.1), and we have already proved in

Theorem 3.3.2 that period-3 orbits are dense, we only need to prove that also the στ-

Bernoulli shift orbits, which are periodic with period T Lσ≤ , are dense. But this can be

26

done by using the same procedure as for Theorem 3.3.2, in which we swap the role of

and . In practice, we can easily prove that given a σ

nB

nP1 τ-Bernoulli shift bit string ( )Tx L

for rule 131 , there exists an integer such that in the noise tail of . m n n>B P1 ( )mLy

Consequently, all periodic orbits of rule 131 are dense, and hence rule 131 is

perfect.

There is an important observation to make about the ω-limit orbits of rule 131 , which

was already mentioned at the end of Part 2 of Theorem 3.2.1. For ,L L∀ ∈ , the bit

string 1 is a period-1 attractor for rule1 1… 9 131 . However, according to the definition

of period-T and στ-Bernoulli shift orbits we gave in [Chua et al., 2005a], such attractor

can also be considered as a period-T orbit, T∀ , and as a στ-Bernoulli shift orbit, ,σ τ∀ .

Here, our choice is considering it either as period-3 or as στ-Bernoulli shift with σ=1 and

τ=1, according to our convenience, so that we do not have to analyze further cases in

Theorem 3.3.2 and 3.3.3. Observe that in Tables 5b and 6b we indicate such orbit as

period-1.

Now, we can focus on the density of the robust ω-limit orbits of rule 133 .

Theorem 3.3.4 Dense period-6 orbits of rule 133

Rule 133 has a dense set of period-6 orbits.

Proof

According to what we proved in Theorem 3.2.2, any finite length periodic bit

string containing at least one ‘wall’ can be only period-1, period-2, period-3, or period-6.

Moreover, we proved that the two ‘walls’ isolate the dynamic behavior of the bit

sequence within such ‘walls’.

9 Table 14(B) of [Chua et al., 2008] shows a list of all rules endowed with Type B perod-1 orbits, which includes rules 131 and 133 .

27

With a procedure similar to the one discussed in the proof of Theorem 3.3.3, we

can affirm that the noise tail of the bit string , which we use to ‘test’ the density of

rule

( )mLy

131 , can be arbitrarily long; hence, we can always create a noise tail delimited by

‘walls’ and containing a period-6 bit string, so that the whole bit string is period-6,

as a consequence of Theorem 3.1.1. 6( ) (mL mL≡y y )

In conclusion, rule 133 has a dense set of period-6 orbits.

We cannot give any definitive result on whether rule 133 is perfect, because our

empirical analysis shows that this rule has an unbounded number of ω-limit orbits of

distinct periods.

Before concluding this section, we would like to introduce yet another

fundamental concept:

Definition 3.3.3 Riddled basins of attraction for rule N

Rule N is said to have riddled basins of attraction iff for any 0ε > and any period-

orbit

T

( )( )T L L∈Σx , for all possible periods T , ∃ an integer such that for all m p m≥

there is a period-T orbit ′ ( ) ( )T pL pL′ ∈Σy , ( )( )T TpL p′ ≠y x L , for all possible periods

, such that T ′ ( ) ( )T TpL pL ε′− <x y .

This definition implies that for any 0ε > (and hence for any position , as defined in

Fig. 40), given any period-T orbit and concatenating it times to generate the bit

string , where , we can create another period-T bit string where

the first t bits of and coincide. Our definition of riddled basins is

inspired by the one given in [Alexander et al., 1992] for continuous systems.

t

( )T Lx m

( )T mLx mL t> ( )T mL′y

( )T mLx ( )T mL′y

A simple example can illustrate the meaning of the notion of riddled basins.

Example 3.3.2

28

Let us consider the bit string ( ) ( ) ( )12 001001111001T TL = =x x , which is period-

3 for rule 131 , as shown in Fig. 41, and let us fix 25 82 3 10ε − −= ≈ ⋅ . Since 131 has

either στ-Bernoulli shift orbits or period-3 orbits (see Theorem 3.2.1), if we suppose that

131 has riddled basins of attraction (as it will indeed be proved in Theorem 3.3.5) then

we should be able to find a bit string ( )T mL′y belonging to a στ-Bernoulli shift orbit

closer than ε to the bit string ( )T mLx obtained by concatenating times . From

the expression of

m ( )T Lx

ε reported above, we find easily that the position of the bit tx (see Fig.

40) is 24x ; consequently, because 3m ≥ 12L = for ( )T Lx . In Table 8, we displayed the

binary and the decimal representations of the bit strings ( )T Lx and , whereas

their space-time patterns are in Figs. 41 and 42, respectively. Now, we need to find a bit

string belonging to a σ

(3T Lx )

)(3T L′y τ-Bernoulli shift orbit which has the first 25 bits of

( )T Lx : we found experimentally that the bit string ( )3T L′y reported in Table 8, and

whose space-time pattern is in Fig. 43, meets these conditions. A visual representation of

this example is given in Fig. 44.

It is not by chance that two bit strings belonging to different kind of ω-limit orbits were

arbitrarily close, because for rule 131 the following theorem holds:

Theorem 3.3.5 Riddled basins of attraction of rule 131

The basins of attraction of rule 131 are riddled.

Proof

It follows directly from the proofs of Theorems 3.3.2 and 3.3.3. From the arbitrariness of

the noise tail in , we conclude that we can change the proportion of patterns

and , and hence the kind of orbit, according to our convenience.

(T mL′y ) nP1

nB

29

We will see in our forthcoming articles that 131 is not the only rule having riddled

basins of attraction.

4. Permutive rules

In [Chua et al., 2008], we described a particular class of CA local rules called

‘permutive’, whose Boolean cubes exhibit on anti-symmetry with respect to some vertical

plane through the center of the cube, as depicted in Fig. 8 of [Chua et al., 2008]. In

particular, we say that a local rule N is: 1) Left-Permutive iff the vertical symmetry

plane is parallel to the paper; 2) Right-Permutive iff the vertical symmetry plane is

perpendicular to the paper; 3) Bi-Permutive, iff it is both Left- and Right-Permutive.

Finally, a local rule N is said to be Permutive iff N is Left and / or Right-Permutive.

An examination of the 256 Boolean cubes in Table 1 of [Chua et al., 2008] shows that

there are only 16 Left-Permutive rules, 16 Right-Permutive rules, and 4 Bi-Permutive

rules, as displayed in Tables 9, 10, and 11, respectively.

The symmetry in the Boolean Cubes of Permutive rules is a consequence of the

symmetry of their truth tables (see Appendix C of [Chua et al., 2008]). We recall that if

N is left-permutive then

1 11 1 1 1( ) ( )t t t t t t t t

i i i i i i i iN NT x x x x T x x x x+ +

− + − += ⇒ =

and if N is right-permutive then

1 11 1 1 1( ) ( )t t t t t t t t

i i i i i i i iN NT x x x x T x x x x+ +

− + − += ⇒ =

for { }1 1, , 0,1i i ix x x− + ∈ . If we use the compact representation [Chua et al., 2002] of a local

rule 7 6 5 4 3 2 1 0( )N β β β β β β β β= , where 7

02i

ii

N β=

= ⋅∑ , we can easily find that a local rule

is left-permutive iff (see also Fig. 8a of [Chua et al., 2008])

7 3β β= , 6 2β β= , 5 1β β= , and 4 0β β= , where { }0,1iβ ∈ (7a)

or, alternatively

30

( )3 2 1 0 3 2 1 0N β β β β β β β β= (7b)

and it is right-permutive iff (see also Fig. 8b of [Chua et al., 2008])

7 6β β= , 5 4β β= , 3 2β β= , and 1 0β β= , where { }0,1iβ ∈ (8a)

or, alternatively

( )6 6 4 4 2 2 0 0N β β β β β β β β= (8b)

In fact, the original definition of Permutive rules [Hedlund, 1929] was given through the

property of the truth tables of the local rules.

Not all the permutive rules are globally-independent, as already observed in

[Chua et al., 2008]. The following theorem allows us to find the relation among left- and

right-permutive rules:

Theorem 4.1 Right-permutivity and Global transformations

Let N be a right-permutive rule, then: 1) ( )† †N T N= is left-permutive; 2)

(N T N= ) is right-permutive; 3) ( )N T N∗ ∗= is left-permutive.

Proof

First, we recall from [Chua et al., 2004] that †7 3 5 1 6 2 4 0( )N β β β β β β β β= . If N is

right-permutive, from Eq. (8a) we obtain †6 2 4 0 6 2 4 0( )N β β β β β β β β= , which is left-

permutive, as follows from Eq. (7b).

Second, we recall from [Chua et al., 2004] that 0 1 2 3 4 5 6 7( )N β β β β β β β β= . If

N is right-permutive, from Eq. (8a) we obtain 0 0 2 2 4 4 6 6( )N β β β β β β β β= which is

right-permutive, as follows from Eq. (8b).

Third, we recall from [Chua et al., 2004] that 0 4 2 6 1 5 3 7( )N β β β β β β β β∗ = . If N

is right-permutive, from Eq. (8a) we obtain 0 4 2 6 0 4 2 6( )N β β β β β β β β∗ = , which is left-

permutive, as follows from Eq. (7b).

31

Remark

Theorem 4.1 can be proved by inspection of the respective Boolean cubes of Figs. 8(a),

8(b), and 8(c), respectively.

A similar theorem holds for left-permutive rules, mutatis mutandis:

Theorem 4.2 Left-Permutivity and Global transformations

Let us N be a left-permutive rule, then: 1) ( )† †N T N= is right-permutive; 2)

(N T N= ) is left-permutive; 3) ( )N T N∗ ∗= is right-permutive.

Table 12 shows the 10 globally-independent permutive minimal rules, as defined in Sec.

1. Permutive rules have interesting properties, especially for bi-infinite bit strings.

Observe that there is a dramatic change between L finite and L infinite. As an example,

we can see how the notion of surjectivity varies in these two cases.

Definition 4.1 Surjective local rule

A local rule N is said to be surjective if given an arbitrary bit string x there always

exists at least another bit string ′x generating under rule x N , N′⎯⎯→x x

We proved in [Chua et al., 2008] that all rules, except for the six lossless rules

15 , 51 , 85 , 170 , 204 , and 240 , have Gardens of Eden for finite L. Therefore, for

L finite, only the lossless rules are surjective. This is not true when L is infinite, since the

following theorem holds:

Theorem 4.2

For , all permutive rules are surjective. L = ∞

Proof

32

It is sufficient to prove that all bit strings have at least one predecessor, or, in other words,

there are no Gardens of Eden.

Here, we give the proof for a right-permutive rule N ; the proof for left-permutive rules

follows easily, mutatis mutandis. We consider an arbitrary bi-infinite bit string , and

we try to find its predecessor starting with two generic bits

nx

1nix − and n

ix : our task is to

show that it is always possible to find 1 1 1 12 1, , ,n n n n

i i i i 1x x x x− − − −− − + which generate the two generic

bits 1nix − and n

ix . From Eq. (8b), it follows that10 { }1 11 , 0,1n n

i ix x− −−∀ ∈ {, }1

1! 0,1nix −+∃ ∈ such

that ( )1 1 11 1

n n n ni i i iNT x ; consequently, x x x− − −− + = 1

1nix −+ depends only on n

ix and it always exists.

As for 12

nix −− , from Eq. (8b) we find that if 1

2nix −−∃ such that ( )1

2 00niN 1

nix−T x − −= then either

( )12 100n n

i iNT x x−

− −= or ( )12 101n n

i iNT x ; if x−

− −= 12

nix −−∃ such that ( )1

2 00niN 1

nix−T x − −= , then both

( ) 1001 niN

T x −= and ( ) 1101 niN

T x −= . This means that for any 1nix − , { }1

2 0,1nix −−∃ ∈ such that

( )1 1 12 1

n n n ni i i iN

T x x x x− − −− − −= 1 if ( ) ( )1 1

1 00n ni ix x− −− = , or ( ) ( )1 1

1 01n ni ix x− −− = . Similar considerations

lead to the conclusion that for any 1nix − , { }1

2 0,1nix −−∃ ∈ such that ( )1 1 1

2 1n n n ni i i iN 1x x x− − −T x − − −= if

, or (( ) ( )1 11 11n n

i ix x− −− = ) ( )1 1

1 10n ni ix x− −− = . Therefore, { }1 1

1 , 0,1n ni ix x− −− {∀ ∈ , }1

2 0,1nix −−∃ ∈ such

that ( )1 1 12 1

n n n ni i i iN 1x x x− − −− − −=

ni

T x . This proves that there is always a predecessor of nx

1,nix x−∀ . Since 1

nix − and n

ix are arbitrary, the proposition holds in general.

Theorem 4.3

For , all bit strings of bi-permutive rules have exactly four predecessors. L = ∞

Proof

We consider an arbitrary bi-infinite bit string , and we try to find its predecessor

starting with two generic bits

nx

1nix − and n

ix : our task is to show that it is always possible to

find 1 1 12 1, , ,n n n n

i i i i1

1x x x x− − − −− − + which generate the two generic bits 1

nix − and n

ix . From Eq. (8b), it

10 We recall that the notation means “there exists only one”. !∃

33

follows that { }1 11 , 0,1n n

i ix x− −−∀ ∈ , { }1

1! 0,1nix −+∃ ∈ such that ( )1 1 1

1 1n n n ni i i iN

T x ;

consequently,

x x x− − −− + =

11

nix −+ depends only on n

ix and it always exists. From Eq. (8b), it follows

that { }1 11 , 0,1n n

i ix x− −−∀ ∈ , { }1

2! 0,1nix −−∃ ∈ such that ( )1 1 1

2 1n n n ni i i iN 1x x x− − −T x − − −= ; consequently,

12

nix −− depends only on 1

nix − and it always exists. From the uniqueness of 1

1nix −+ and 1

1nix −− , and

the arbitrariness of 11

nix −− and 1n

ix − , the proposition follows.

Remark

The results of Theorems 4.2 and 4.3 are not new, since they can be found also in the

classical works of the CA literature, such as [Hedlund, 1929] and [Wolfram, 2002].

However, here we presented compact and straightforward proofs for them.

5. Concluding remarks In this paper, we have introduced several fundamental notions of our ‘Nonlinear

Dynamics Perspective of Cellular Automata’. For example, we have presented a new

criterion for selecting 88 globally-independent local rules (and the 82 quasi globally-

independent local rules) as our prototype, which has the great advantage of being the

result of a rigorous choice. Our choice also allows us to train our monkey mascot to learn

only the smallest number of firing patterns for each prototype rule. In this new listing, the

“universal Turing machine” rule 137 , which was extensively cited in our last two papers,

is listed as one of the 88 globally-independent local rules.

The main part of this paper is concerned with the two rules of Group 3; namely,

131 and 133 . Their dynamical behaviors are unique among the 88 globally-

independent local rules: rule 131 has robust period-3 ω-limit orbits and non-robust στ-

Bernoulli shift ω-limit orbits, whereas rule 133 has robust period-6 ω-limit orbits and

non-robust period-T ω-limit orbits with T 6≠ . Here, for the first time, we found these

results through a rigorous procedure and not by means of computer experiments. We also

defined the notion of dense set of period-T orbits, which is the starting point to introduce

more complex concepts such as the perfect period-T orbit sets and the riddled basins of

34

attractions. For instance, we proved that all ω-limit orbits of 131 are dense and hence

131 is perfect, in this precise sense, and that the basins of attraction of 131 are riddled.

We conjecture that similar results can be found for many of the rules of Groups 1 and 2

as well. We emphasize the importance of the Noise-tail Bounding Estimate (Proposition

3.3.1) in our discourse, since it allows us to introduce a metric for binary bit strings.

Finally, we have given a few results on permutive rules, which were defined in

our previous papers but never studied thoroughly. There are only 10 globally-independent

permutive rules: 7 of them are right- / left- permutive, and 3 of them are bi-permutive.

Observe that in Theorem 4.1 we have proved that the distinction between right- and left-

permutive rules is fictitious, since every right-permutive has a globally-equivalent left-

permutive rule, and vice versa. Also, we have shown a straightforward proof of two well-

known results regarding the predecessors of permutive rules.

This paper can be considered as a sort of ‘bridge’ between our previous works and

our future ones: the table of the 88 globally-independent rules is now cast in stone and

properly justified, both from our scientific and neural perspective; for the first time, we

were able to characterize completely the ω-limit orbits, robust and non-robust, of one

local rule, 131 ; our results on the surjectivity of permutive rules show how different the

same rule can behave for finite or bi-infinite bit strings. All these elements are the

cornerstones of our future discourse.

Acknowledgement This work is supported in part by the Hungarian Academy of Science and the USA Office

of Naval Research under grant no. N000 14-09-1-0411. The authors thank Dr. J. Shin for

drawing the basin tree diagrams and the phase portraits.

Appendix A Group 5 (Hyper Bernoulli rules) contains ten globally-independent local rules.

Two of them have been renumbered according to the criterion illustrated in Sec. 1:

namely, 129 substitutes 126 and 161 substitutes 122 . The basin tree diagrams and

35

the portraits of the ω-limit orbits of these two rules are displayed in Tables A1, A2, A3,

and A4, respectively.

Appendix B Group 6 (Complex Bernoulli rules) contains eight globally-independent local

rules. Two of them have been renumbered according to the criterion illustrated in Sec. 1:

namely, 137 substitutes 110 and 166 substitutes 154 . The basin tree diagrams and

the portraits of the ω-limit orbits of these two rules are displayed in Tables B1, B2, B3,

and B4, respectively.

Appendix C The basin tree diagrams of the rules belonging to Group 5 (Hyper Bernoulli rules)

were displayed in [Chua et al., 2007a]. In Tables C1-C8, we display the portraits of the

ω-limit orbits of such rules, except for 129 and 161 , which can be found in Appendix

A.

Appendix D The basin tree diagrams of the rules belonging to Group 6 (Complex Bernoulli

rules) were displayed in [Chua et al., 2007a]. In Tables D1-D6, we display the portraits

of the ω-limit orbits of such rules, except for 137 and 166 , which can be found in

Appendix B.

36